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When I was studying for my chemistry test, I found an equation that illustrates how to get water gas (carbon monoxide + hydrogen) out of methane. It is stated like this: $$\begin{align}\ce{2CH4 + CO2 + H2O &->[\space \Delta ] 3CO +5H2}\tag{1}\\ \end{align}$$

First, I tried to balance it by myself in order not to memorize it but I ended with this equation: $$\begin{align}\ce{3CH4 +CO2 +2H2O &->[\space \Delta] 4CO +8H2}\tag{2}\\ \end{align}$$ Then, I noticed that if we make the coefficient Of CO2 one, we can get infinite number of ways to balance the equation by changing the coefficient of CH4

We can make a third trial: $$\begin{align}\ce{4CH4 +CO2 +3H2O &->[\space \Delta] 5CO +11H2}\tag{3}\\ \end{align}$$ Have you seen the pattern? I can conclude my observations in this equation: $$\begin{align}\ce{nCH4 +CO2 +(n -1)H2O &->[\space \Delta] (n +1)CO +(3n -1)H2}\tag{4}\\ \end{align}$$ I am confused. Which one is true? If all of them are true, why my textbook mentions only one of them? What really happens in reality? Is equation $2$ just a simplification of more complicated equations? Does this have to do anything with the reaction mechanism?

I need to get more intuition about what is happening and the causes of this special feature.

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We first note that your reaction (4) can be broken down into two reactions by splitting up those chemical species that depend on $n$ and those that don't: $$\begin{align}\ce{CH4 + H2O &-> CO + 3H2}\\ \ce{CO2 + H2 &-> CO + H2O}, \end{align}$$ so your reactions (1) through (3) are just statements about the relative rates of these two reactions.

Which one is true? If all of them are true, why my textbook mentions only one of them?

I presume none of them are really true, in the sense that the ratio of the reaction rates isn't something as nice as 1:1 or 1:2 or 1:3, etc., and we haven't accounted for side reactions either. But assuming a 1:1 ratio is certainly a useful simplification.

What really happens in reality?

The ratio of the rates will depend on the temperature and pressure and composition of the feed. There will be side reactions that occur. You can perform experimental measurements of the product composition to determine what happens.

Is equation 2 just a simplification of more complicated equations?

Yes.

Does this have to do anything with the reaction mechanism?

No. The overall reaction does not, in general, tell you anything about the reaction mechanism.

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    $\begingroup$ The more basic equations are: $$\ce{CH4 + H2O <=> CO + 3H2}$$ $$\ce{CH4 + CO2 <=> 2CO + 2H2}$$ $\endgroup$ – MaxW Aug 21 '18 at 22:59
  • $\begingroup$ After reading your answer, I think the general basic equations should be: $$\begin{align}\ce{(n-1)CH4 +(n-1)H2 &-> (n-1)CO +(3n-3)H2}\tag{5}\\\end{align}$$ $$\begin{align}\ce{CH4 +CO2 &-> 2CO +2H2}\tag{6}\\\end{align}$$ If we add those two equations, we get equation 4 and each one of them has only one way to balance it. Thank you for this illustration. $\endgroup$ – QEDady Aug 22 '18 at 1:35
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    $\begingroup$ @AmerEsmail Your first one is just $(n - 1)$ times CH_4 + H_2 -> CO + H_2 $\endgroup$ – wizzwizz4 Aug 22 '18 at 6:20

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